Skip to main content
Springer Nature Link
Log in

Highly Efficient and Energy Stable Schemes for the 2D/3D Diffuse Interface Model of Two-Phase Magnetohydrodynamics

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we consider a two-phase MHD (magnetohydrodynamics) model based on the phase-field method. The governing model consists of the phase field equations, the Navier–Stokes equations and the Maxwell equations via the convection term, surface stress force, the Lorentz force and Ohm’s law. The model couples the phase field function, velocity, magnetic field, pressure and the introduced Lagrange multiplier, which results in a strong coupled, nonlinear and double saddle points type system. To solve the model efficiently, we propose two totally decoupled, linear and unconditionally energy stable schemes. The given schemes combine semi-implicit stabilization/IEQ methods for the phase-field equations, projection method for the double saddle points MHD equations (Schötzau in Numer Math 96:771-800, 2004) with some subtle implicit-explicit treatments for nonlinear coupled terms. For space discretization, the magnetic field is approximated by the \(H(\mathrm {curl})\)-conforming edge element. Then, the coupled, nonlinear and saddle point type model is separated into a series of small elliptic type problems. We also rigorously prove the unconditional energy stabilities of the proposed schemes in both temporal discretization and full discretization cases. To our knowledge, it is the first totally decoupled and unconditionally stable scheme for the two-phase MHD problem. Finally, we implement some numerical examples, including accuracy tests, energy stability tests, 2D and 3D two-phase electromagnetic driven cavity benchmark problems, to verify the stability and convergence of our schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Schötzau, D.: Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96, 771–800 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Layton, W., Lenferink, H., Peterson, J.: A two-level Newton finite element algorithm for approximating electrically conducting incompressible fuid flows. Comput. Math. Appl. 28(5), 21–31 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Layton, W., Meir, A., Schmidtz, P.: A two-level discretization method for the stationary MHD equations. Electron. Trans. Numer. Anal. 6, 198–210 (1997)

    MathSciNet  MATH  Google Scholar 

  4. Liu, J., Pego, R.: Stable discretization of magnetohydrodynamics in bounded domains. Commun. Math. Sci. 8, 235–251 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Li, X., Zheng, W.: A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D. J. Comput. Phys. 351, 254–270 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zhang, G.D., Zhang, Y., He, Y.: Two-level coupled and decoupled parallel correction methods for stationary incompressible magnetohydrodynamics. J. Sci. Comput. 65, 920–939 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhang, G.D., He, Y.: Decoupled schemes for unsteady MHD equations II: finite element spatial discretization and numerical implementation. Comput. Math. Appl. 69, 1390–1406 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhang, G.D., He, Y.: Decoupled schemes for unsteady mhd equations I: time discretization. Numer. Methods Partial Differ. Equ. 33, 956–973 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhang, G.D., Yang, X., He, X.: A decoupled, linear and unconditionally energy stable scheme with finite element discretizations for magneto-hydrodynamic equations. J. Sci. Compt. 81, 1678–1711 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Su, H., Feng, X., Huang, P.: Iterative methods in penalty finite element discretization for the steady MHD equations. Comput. Method Appl. Mech. Eng. 304, 521–545 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Su, H., Feng, X., Zhao, J.: Two-level penalty Newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics equations. J. Sci. Comput. 70(3), 1144–1179 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shahri, M., Sarhaddi, F.: Second law analysis for two-immiscible fluids inside an inclined channel in the presence of a uniform magnetic field and different types of nanoparticles. J. Mech. 34(4), 541–549 (2018)

    Article  Google Scholar 

  13. Xie, Z., Jian, Y.: Entropy generation of magnetohydrodynamic electroosmotic flow in two-layer systems with a layer of non-conducting viscoelastic fluid. Int. J. Heat Mass Trans. 127, 600–615 (2018)

    Article  Google Scholar 

  14. Hadidi, A., Jalali-Vahid, D.: Numerical simulation of dielectric bubbles coalescence under the effects of uniform magnetic field. Theor. Comput. Fluid Dyn. 30, 165–184 (2016)

    Article  Google Scholar 

  15. Thome, R.: Effect of a transverse magnetic field on vertical two-phase flow through a rectangular channel, Argonne National Laboratory Report, No. ANL-6854 (1964)

  16. Ki, H.: Level set method for two-phase incompressible flows under magnetic fields. Comput. Phys. Commun. 181, 999–1007 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ansari, M., Hadid, A., Nimvari, M.: Effect of a uniform magnetic field on dielectric two-phase bubbly flows using the level set method. J. Mag. Mag. Mater. 324, 4094–4101 (2012)

    Article  Google Scholar 

  18. Rayleigh, L.: On the theory of surface forces.- II Compressible fluids. Philols. Mag. 33(201), 209–220 (1892)

    Article  MATH  Google Scholar 

  19. van der Waals, J.D.: The thermodynamic theory of capillarity under the hypothesis of a continuous density variation. J. Stat. Phys. 20, 197–244 (1893)

    Google Scholar 

  20. Anderson, D., McFadden, G., Wheeler, A.: Diffuse-interface methods in fluid mechanics. Ann. Rev. Fluid Mech. 30, 139–165 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gurtin, M., Polignone, D., Vinals, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6(6), 815–831 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jacqmin, D.: Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 155(1), 96–127 (2007)

    Article  MathSciNet  Google Scholar 

  23. Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D. 179(3–4), 211–228 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A. 454, 2617–2654 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yue, P., Feng, J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang, J., Mao, S., He, X., Yang, X., He, Y.: A diffuse interface model and semi-implicit energy stable finite element method for two-phase magnetohydrodynamic flows. Comput. Methods Appl. Mech. Eng. 356, 435–464 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Costabel, M., Dauge, M.: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93, 239–277 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gao, H., Qiu, W.: A linearized energy preserving finite element method for the dynamical incompressible magnetohydrodynamics equations. arXiv:1801.01252 [math.NA]

  29. Greif, C., Li, D., Schötzau, D., Wei, X.: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Appl. Mech. Eng. Comput. Method 199, 45–48 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Houston, P., Schötzau, D., Wei, X.: A mixed DG method for linearized incompressible magnetohydrodynamics. J. Sci. Comput. 40, 281–314 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Choi, H., Shen, J.: Efficient splitting schemes for magneto-hydrodynamic equations. Sci. China Math. 59, 1495–1510 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, X., Zhang, G.D., He, X.: Convergence analysis of an unconditionally energy stable projection scheme for magneto-hydrodynamic equations. Appl. Numer. Math. 136, 235–256 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Guermond, J., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shen, J., Yang, X.: Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53(1), 279–296 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhu, J., Chen, L., Shen, J., Tikare, V.: Morphological evolution during phase separation and coarsening with strong inhomogeneous elasticity. Model. Simul. Mater. Sci. Eng. 9, 499–511 (2001)

    Article  Google Scholar 

  36. Xu, C., Tang, T.: Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44, 759–1779 (2006)

    Article  MathSciNet  Google Scholar 

  37. Yang, X.: Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete Contin. Dyn. Syst. Ser. B 11, 1057–1070 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. 28, 1169–1691 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Chen, C., Yang, X.: Efficient numerical scheme for a dendritic solidification phase field model with melt convection. J. Comput. Phys. 388, 41–62 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. Chen, C., Yang, X.: Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn–Hilliard model. Comput. Method Appl. Mech. Eng. 351, 35–59 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Chen, R., Yang, X., Zhang, H.: Second order, linear, and unconditionally energy stable schemes for a hydrodynamic model of smectic-A liquid crystals. SIAM J. Sci. Comput. 39(6), A2808–A2833 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294–316 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yang, X.: Efficient linear, stabilized, second-order time marching schemes for an anisotropic phase field dendritic crystal growth model, Comput. Method. Appl. Mech. Eng. 347, 316–339 (2019)

    Article  MATH  Google Scholar 

  44. Yang, X., Ju, L.: Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model. Comput. Method Appl. Mech. Eng. 315, 691–712 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  45. Yang, X., Yu, H.: Efficient second order unconditionally stable schemes for a phase field moving contact line model using an invariant energy quadratization approach. SIAM J. Sci. Comput. 40(3), B889–B914 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  46. Yang, X., Zhao, J., He, X.: Linear, second order and unconditionally energy stable schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method. J. Comput. Appl. Math. 343, 80–97 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  47. Yang, X., Zhao, J., Wang, Q., Shen, J.: Numerical approximations for a three-component Cahn-Hilliard phase-field model based on the invariant energy quadratization method. Math. Mod. Meth. Appl. S. 27(11), 1993–2030 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Yang, X., Zhang, G.D.: Convergence analysis for the invariant energy quadratization (IEQ) schemes for solving the Cahn–Hilliard and Allen–Cahn equations with general nonlinear potential. J. Sci. Comput. 82(3), 1–28 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zhang, X.: Sharp-interface limits of the diffuse interface model for two-phase inductionless magnetohydrodynamic fluids. arXiv:2106.10433 [math.AP]

  50. Lin, F., Liu, C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48(5), 501–537 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  51. Nochetto, R., Salgado, A., Ignacio, T.: A diffuse interface model for two-phase ferrofluid flows. Comput. Method Appl. Mech. Eng. 309, 497–531 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  52. Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst 28(4), 1669–1691 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  53. Shen, J., Yang, X.: Energy stable schemes for Cahn–Hilliard phase-field model of two-phase incompressible flows. Chin. Ann. Math. B 31(5), 743–758 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  54. Girault, V., Raviart, P.: Finite Element Method for Navier–Stokes Wquations: Theory and Algorithms, pp. 395–414. Springer, Berlin (1987)

    Google Scholar 

  55. Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis, 343 (2001), American Mathematical Soc

  56. Kay, D., Styles, V., Welford, R.: Finite element approximation of a Cahn–Hilliard–Navier–Stokes system. Interface. Free Bound. 10, 15–43 (2008)

    Article  MathSciNet  Google Scholar 

  57. Hintermüller, M., Hinze, M., Kahle, C.: An adaptive finite element Moreau–Yosida-based solver for a coupled Cahn–Hilliard/Navier–Stokes system. J. Comput. Phys. 235, 810–827 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  58. Boyer, F.: A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31, 41–68 (2002)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Haiyan Su

  2. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, Shandong, People’s Republic of China

    Guo-Dong Zhang

Authors
  1. Haiyan Su
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Guo-Dong Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Guo-Dong Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is in part supported by the NSF of China (Grant No. 11701493, 11771375, 12171415) and Tianshan Youth Project of Xinjiang Province (No. 2017Q079).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Su, H., Zhang, GD. Highly Efficient and Energy Stable Schemes for the 2D/3D Diffuse Interface Model of Two-Phase Magnetohydrodynamics. J Sci Comput 90, 63 (2022). https://doi.org/10.1007/s10915-021-01741-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01741-3

Keywords